Optimal Control Problem Research Articles

Distributed virtual formation control for railway trains with nonlinear dynamics and collision avoidance constraints

To improve the model accuracy and control efficiency for the movements of a virtual formation, this paper investigates distributed optimal control for the virtual formation control system in railways. Adopting the relative distance braking mode, a coupled optimal control problem with nonlinear train dynamics and constraints regarding collision avoidance and jerk is formulated for the virtual formation. To handle the non-convex constrained problem efficiently, a distributed augmented Lagrangian based alternating direction inexact newton (ALADIN) method under the model predictive control (MPC) framework is developed. For the execution of the distributed computational process, the copied variables are introduced to reformulate the original coupled problem in an objective separable form. By exploiting the problem separability, the ALADIN method decomposes the reformulation into a coordinated quadratic programming problem of small-scale and several local nonlinear programming problems that can be calculated in parallel, thereby facilitating real-time control and relieving communication burden. Numerical experiments on a metro line are carried out to verify the effectiveness of the proposed model and method. Experimental results demonstrate that high-performance tracking control for virtually coupled train units can be achieved in real time.

Low rank approximation method for perturbed linear systems with applications to elliptic type stochastic PDEs

In this paper, we propose a low rank approximation method for efficiently solving stochastic partial differential equations. Specifically, our method utilizes a novel low rank approximation of the stiffness matrices, which can significantly reduce the computational load and storage requirements associated with matrix inversion without losing accuracy. To demonstrate the versatility and applicability of our method, we apply it to address two crucial uncertainty quantification problems: stochastic elliptic equations and optimal control problems governed by stochastic elliptic PDE constraints. Based on varying dimension reduction ratios, our algorithm exhibits the capability to yield a high-precision numerical solution for stochastic partial differential equations, or provides a rough representation of the exact solutions as a pre-processing phase. Meanwhile, our algorithm for solving stochastic optimal control problems allows a diverse range of gradient-based unconstrained optimization methods, rendering it particularly appealing for computationally intensive large-scale problems. Numerical experiments are conducted and the results provide strong validation of the feasibility and effectiveness of our algorithm.

Adaptive optimal concurrent control for detumbling space non-cooperative target via multipoint repeated contact

Space debris generally exhibits complex tumbling motions, and its direct capture may cause damage to space manipulator and base spacecraft. Current detumbling strategies typically require continuous contact collisions with the target and do not take into account actuator limitations. Thus, this paper presents an adaptive optimal concurrent control of space free-floating multi-fingered robot (SFMR) for multipoint repeated contact detumbling of non-cooperative targets. Firstly, the multipoint repeated intermittent contact model between the SFMR and the target is established. Further, an optimal admittance control that considers manipulator actuator limits and target motion bounds is formulated to generate a compliant detumbling trajectory. Through transforming the state and input inequality constraints into extended dynamical subsystems and saturation functions, respectively, the optimal control problem (OCP) is transformed into a readily solvable equality constraint problem. Moreover, an enhanced nonsingular terminal sliding mode (ENTSM) control with radial basis function neural network (RBFNN) compensation is presented in the presence of lumped uncertainties and external disturbances, which achieves rapid finite-time convergence and low-chattering. The simulation results show that the proposed method can reduce the velocity of the space target effectively without causing base spacecraft interference while achieving accurate trajectory tracking.

Dynamic event-driven optimal consensus control for state-constrained multiagent zero-sum differential graphical games

In this paper, a dynamic event-driven optimal control scheme is proposed for the zero-sum differential graphical games in nonlinear multiagent systems with full-state constraints. Initially, to address the dual demands of optimality and state constraints, a set of system transformation functions are introduced to satisfy the state constraints of the agents. Then, by applying the principle of differential game theory, the distributed optimal control problem affected by external disturbances is formulated as a zero-sum differential graphical game, and the performance index function related to neighbor informations and disturbances is designed for each follower. Afterwards, to enhance the utilization of communication resource, a novel dynamic event-triggered mechanism characterized by a dynamic threshold parameter and an auxiliary dynamic variable is developed, which not only exhibits greater flexibility but also diminishes the frequency of triggers. Furthermore, the approximate optimal control strategies are obtained by employing an event-driven adaptive dynamic programming algorithm. Ultimately, a simulation example is presented to verify the applicability of the proposed control approach.

Optimal control for linear cyber physical systems under replay attacks via time delay estimation approach

We mainly study the optimal control problem for linear cyber physical systems under replay attacks in this paper. The methodological research mainly relies on the following two starting points: (1) Promptly detect and estimate the malicious replay attack behaviours hampering normal system procedures; (2) Design a resilient optimal control strategy to ensure the system regularly works as much as possible. Specifically, the linear cyber physical system becomes a time delay system after the replay attack, and then an algorithm is designed to transform the system into a delay-free system and ensure its stability. Finally, a numerical simulation is employed to demonstrate the practicality and efficacy of the proposed approach.

Effect of awareness and saturated treatment on the transmission of infectious diseases

Abstract In this article, we study the role of awareness and its impact on the control of infectious diseases. We analyze a susceptible-infected-recovered model with a media awareness compartment. We find the effective reproduction number R 0 {R}_{0} . We observe that our model exhibits transcritical forward bifurcation at R 0 = 1 {R}_{0}=1 . We also performed the sensitivity analysis to determine the sensitivity of parameters of the effective reproduction number R 0 {R}_{0} . In addition, we study the corresponding optimal control problem by considering control in media awareness and treatment. Our studies conclude that we can reduce the rate of spread of infection in the population by increasing the treatment rate along with media awareness.

Numerical solution of optimal control problems for linear systems of ordinary differential equations

Abstract An original numerical matrix algorithm aimed at solving the optimal control problems for linear systems of ordinary differential equations with constant coefficients is proposed. The work of the algorithm is demonstrated with the problem, which consists in generating a given small disturbance of the Poiseuille flow in an infinite duct by blowing and suction through the walls. The costs of creating the leading modes and optimal disturbances are compared, which is of independent interest.

Mathematical analysis of a non‐convex optimal control problem for age‐structured mosquito populations

We present a rigorous mathematical analysis of a non‐convex optimal control problem for mosquito populations. The nonlinear model for the dynamics of the mosquito population takes in consideration the iterations among the immature (aquatic) subpopulation, the adult winged subpopulation, and the environment resources; the immature subpopulation is assumed to be age‐structured. Moreover, the action of certain control mechanisms on these subpopulations is also taken in account. The cost functional to be minimized is non‐convex. The proof of the existence of an optimal control is done by using fixed point arguments and a special minimizing sequence obtained with the help of Ekeland's variational principle.

Solving optimal control problems governed by nonlinear PDEs using a multilevel method based on an artificial neural network

Solving optimal control problems governed by nonlinear PDEs using a multilevel method based on an artificial neural network

A recursive framework of vehicle trajectory planning at mixed‐traffic signalized intersections

AbstractThis study aims to introduce a new strategy for anticipating the behaviour of human‐driven vehicles (HDVs) and designing trajectories for connected and automated vehicles (CAVs) at signalized intersections under mixed traffic scenarios. To tackle the challenge of unreliable HDV trajectory predictions stemming from driving unpredictability, a recursive framework is developed. This framework integrates real‐time tracking data from both traffic detectors and CAVs, continuously updating HDV predictions. The proposed approach employs the updated predictions to formulate optimal control problems recursively to optimize or adjust CAV trajectories, enhancing travel and energy efficiency. Besides, the recomputing of CAV trajectories will only be conducted when the variation in predictions rises to a certain threshold, balancing efficiency and computing consumption, inspired and modified based on MPC methods. The application of the Pontryagin maximum principle aids in finding solutions efficiently by transforming necessary conditions into a system of equations and consolidating elementary unconstrained and constrained arcs. Numerical simulations were carried out to evaluate the performance of the proposed recursive framework, revealing its superiority over the one‐time approach, particularly in isolated intersections with high traffic demands. Additionally, the recursive framework exhibited more robust and effective enhancements throughout the road network.

Stochastic optimal control problem of consumption and pension insurance with uncertain lifetime and its application to real-life data

We consider a continuous-time model of optimal consumption and pension insurance for a consumer with an uncertain lifetime. In the model, the consumer earns a stochastic wage income during her working life and optimally allocates her income between personal consumption, pension insurance, and securities with a deterministic dynamic return. Due to the weak development of the stock market in developing countries, employees' income comes mainly from wages and interest on savings from banks, that are discussed in this paper. The consumer's utility and bequest functions are constant absolute risk aversion (CARA). By characterizing the optimality condition of the consumer's problem using the Hamilton-Jacobi-Bellman equation, we find the optimal consumption and pension insurance as a function of wealth in closed form. We consider an application of the model while estimating its key elements using real-life data on age-specific population size, labor income, and interest rates. We show that as the absolute risk aversion for consumption increases, consumption and wealth move in the opposite direction. We also present a novel finding that wealth and consumption can be negatively related across consumers with different levels of consumption risk aversion.

An efficient ADAM-type algorithm with finite elements discretization technique for random elliptic optimal control problems

We consider an optimal control problem governed by an elliptic partial differential equation (PDE) with random coefficient, and introduce an efficient numerical method for the problem under concern. Based on a finite element discretization of the constraint PDE and objective functional, an ADAM-type iteration scheme is proposed to compute the approximated optimal control, which equips the stochastic gradient iteration with modified momentum acceleration and adaptive step size. We provide theoretical bounds on both the discretization and iteration error of the proposed numerical method. Our method is tested on two examples, and the numerical results show the efficiency and performance of the method.

Fractional-order differential inclusion with pseudo-Lipschitz multi-valued map

In this paper, we investigate the existence of a solution for a fractional-order differential inclusion when the multi-valued map is supposed to be non-convex and satisfies a pseudo-Lipschitz property. For our aim, we use Filippov's theorem. An application to an optimal control problem is also presented for a fractional differential equation.

Numerical solution of different kinds of fractional‐order optimal control problems using generalized Lucas wavelets and the least squares method

Numerical solution of different kinds of fractional‐order optimal control problems using generalized Lucas wavelets and the least squares method

Connecting stochastic optimal control and reinforcement learning

In this paper the connection between stochastic optimal control and reinforcement learning is investigated. Our main motivation is to apply importance sampling to sampling rare events which can be reformulated as an optimal control problem. By using a parameterised approach the optimal control problem becomes a stochastic optimization problem which still raises some open questions regarding how to tackle the scalability to high-dimensional problems and how to deal with the intrinsic metastability of the system. To explore new methods we link the optimal control problem to reinforcement learning since both share the same underlying framework, namely a Markov Decision Process (MDP). For the optimal control problem we show how the MDP can be formulated. In addition we discuss how the stochastic optimal control problem can be interpreted in the framework of reinforcement learning. At the end of the article we present the application of two different reinforcement learning algorithms to the optimal control problem and a comparison of the advantages and disadvantages of the two algorithms.

Extensibility of Solutions of Nonautonomous Systems of Quadratic Differential Equations and Their Application in Optimal Control Problems

Extensibility of Solutions of Nonautonomous Systems of Quadratic Differential Equations and Their Application in Optimal Control Problems

Intelligent-Critic-Based Tracking Control of Discrete-Time Input-Affine Systems and Approximation Error Analysis With Application Verification.

In recent years, the application of function approximators, such as neural networks and polynomials, has ushered in a new stage of development in solving optimal control problems. However, considering the existence of approximation errors, the stability of the controlled system cannot be guaranteed. Therefore, in view of the prevalence of approximation errors, we investigate optimal tracking control problems for discrete-time systems. First, a novel value function is introduced into the intelligent critic framework. Second, an implicit method is utilized to demonstrate the boundedness of the iterative value functions with approximation errors. An explicit method is applied to prove the stability of the system with approximation errors. Furthermore, an evolving policy is designed to iteratively tackle the optimal tracking control problem and demonstrate the stability of the system. Finally, the effectiveness of the developed method is verified through numerical as well as practical examples.

Unraveling the importance of early awareness strategy on the dynamics of drug addiction using mathematical modeling approach.

A drug is any substance capable of altering the functioning of a person's body and mind. In this paper, a deterministic nonlinear model was adapted to investigate the behavior of drug abuse and addiction that incorporates intervention in the form of awareness and rehabilitation. In the mathematical analysis part, the positivity and boundedness of the solution and the existence of drug equilibria have been ascertained, which shows that the model consists of two equilibria: a drug-free equilibrium and a drug endemic equilibrium point. The drug-free equilibrium was found to be both globally and locally asymptotically stable if the effective reproduction number is less than or equal to one (Rc≤1). Furthermore, we were able to show the existence of a unique drug endemic equilibrium whenever Rc>1. Global asymptotic stability of a drug endemic equilibrium point has been ascertained using a nonlinear Lyapunov function of Go-Volterra type, which reveals that the drug endemic equilibrium point is globally asymptotically stable if an effective reproduction number is greater than unity and if there is an absence of a reversion rate of mended individuals (i.e., ω=0). In addition, an optimal control problem was formulated to investigate the optimal strategy for curtailing the spread of the behavior using control variables. The control variables are massive awareness and rehabilitation intervention of both public and secret addicted individuals. The optimal control simulation shows that massive awareness control is the best to control drug addiction in a society. In sensitivity analysis section, the proportion of those who are exposed publicly shows to be a must sensitive parameter that can reduce the reproduction number, and the effective contact rate shows to be a must sensitive parameter to increase the reproduction number. Numerical simulations reveal that the awareness rate of exposed publicly and the rehabilitation rate of addicted publicly are very important parameters to control drug addiction in a society.

Endo-atmospheric maneuver penetration strategy based on generative adversarial reinforcement learning

An intelligent endo-atmospheric penetration strategy based on generative adversarial reinforcement learning is proposed in this manuscript. Firstly, attack and defense adversarial models are established, and missile maneuver penetration problem is transformed into an optimal control problem, considering penetration, handover position and mid-terminal guidance velocity constraints. Then, Radau Pseudospectral method is adopted to generate data samples considering random perturbations. Furthermore, Generative Adversarial Imitation Learning Combined with Deep Deterministic Policy Gradient method (GAIL-DDPG) is designed, with internal process reward signals constructed to tackle long-term sparse reward in missile manuver penetration problem. Finally, penetration strategy is trained and verified. Simulation shows that using generative adversarial reinforcement learning, with sample library to learn expert experience in training early stage, the proposed method can quickly converge. Also, performance is further optimized with reinforcement learning exploration strategy in the later stage of training. Simulation shows that the proposed method has better engineering application ability compared with traditional reinforcement learning method.

Learning and Controlling Multiscale Dynamics in Spiking Neural Networks Using Recursive Least Square Modifications.

Invasive brain-computer interfaces (BCIs) have the capability to simultaneously record discrete signals across multiple scales, but how to effectively process and analyze these potentially related signals remains an open challenge. This article introduces an innovative approach that merges modern control theory with spiking neural networks (SNNs) to bridge the gap among multiscale discrete information. Specifically, the macroscopic point-to-point trajectory is formulated as an optimal control problem with fixed terminal time and state, and it is iteratively solved using the direct dynamic programming (DDP) algorithm. Additionally, SNN is utilized to simulate microscale neural activities in the premotor cortex, employing the product of the weighted adjacency matrix and the mesoscale firing rate to approximate the macroscopic trajectory. The error between actual macroscale behavior and the preceding approximation is then used to update the weighted adjacency matrix through the recursive least square (RLS) method. Analysis and simulation of various tasks, including low-dimensional point-to-point tasks, high-dimensional complex Lorenz systems, and center-out-and-back tasks, verify the feasibility and interpretability of our method in processing multiscale signals ranging from spiking neurons to motion trajectory through the integration of SNN and control theory.